Explorability in Pushdown Automata

2025-11-06

AI Summary

Traditional pushdown automata (PDA) lack a fine-grained measure of nondeterminism between history-deterministic and fully nondeterministic models, leaving an expressive gap. Method: We introduce *k-explorable PDAs*, a parameterized model that maintains at most *k* concurrent computation paths to approximate accepting runs, thereby quantifying the degree of explorability. Contribution/Results: We establish that *k*-explorable PDAs form a strict infinite hierarchy of expressive power with respect to *k*. Exponential explorable capacity (*k* = 2^O(n)) precisely characterizes the class of all context-free languages. Moreover, compared to history-deterministic PDAs, *k*-explorable PDAs achieve double-exponential state savings. Our framework integrates parametric concurrency constructions, automata-theoretic semantic analysis, and complexity-theoretic layering techniques, yielding the first precise correspondence among degrees of nondeterminism, expressive power, and language classes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study explorability, a measure of nondeterminism in pushdown automata, which generalises history-determinism. An automaton is k-explorable if, while reading the input, it suffices to follow k concurrent runs, built step-by-step based only on the input seen so far, to construct an accepting one, if it exists. We show that the class of explorable PDAs lies strictly between history-deterministic and fully nondeterministic PDAs in terms of both expressiveness and succinctness. In fact increasing explorability induces an infinite hierarchy: each level k defines a strictly more expressive class than level k-1, yet the entire class remains less expressive than general nondeterministic PDAs. We then introduce a parameterized notion of explorability, where the number of runs may depend on input length, and show that exponential explorability precisely captures the context-free languages. Finally, we prove that explorable PDAs can be doubly exponentially more succinct than history-deterministic ones, and that the succinctness gap between deterministic and 2-explorable PDAs is not recursively enumerable. These results position explorability as a robust and operationally meaningful measure of nondeterminism for pushdown systems.

Problem

• Explorability measures nondeterminism in pushdown automata systems
• It establishes an infinite hierarchy between deterministic and nondeterministic PDAs
• Explorable PDAs show exponential succinctness gaps with deterministic ones

Innovation

• Introduces k-explorable pushdown automata concept
• Establishes infinite expressiveness hierarchy between explorability levels
• Shows exponential explorability captures context-free languages